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A critical point for random graphs with a given degree sequence. (English) Zbl 0823.05050
Let \(\lambda_ 0,\lambda_ 1,\dots\) be a sequence of non-negative real numbers summing to 1. This paper considers random graphs with \(n\) vertices and approximately \(\lambda_ i n\) vertices of degree \(i\). Let \(Q\) be the sum of the quantities \(i(i- 2)\lambda_ i\) over \(i= 1,2,\dots\). It is shown (under some restrictions) that if \(Q> 0\) then with probability \(\to 1\) as \(n\to \infty\), such graphs have a ‘giant component’, whereas if \(Q< 0\) then all components are small. This result and the proof method yield new insight on the classical ‘double-jump’ threshold for usual random graphs, and can be applied in investigations concerning the chromatic number of sparse random graphs.

MSC:
05C80 Random graphs (graph-theoretic aspects)
60C05 Combinatorial probability
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